Time-harmonic diffuse optical tomography: Hölder stability of the derivatives of the optical properties of a medium at the boundary

We address the inverse problem in Optical Tomography of stably determining the optical properties of an anisotropic medium Ω ⊂ Rn, with n ≥ 3, under the so-called diffusion approximation. Assuming that the scattering coefficient μs is known, we prove H¨older stability of the derivatives of any order of the absorption coefficient μa at the boundary ∂Ω in terms of the measurements, in the time-harmonic case, where the anisotropic medium Ω is interrogated with an input field that is modulated with a fixed harmonic frequency ω = k c , where c is the speed of light and k is the wave number. The stability estimates are established under suitable conditions that include a range of variability for k and they rely on the construction of singular solutions of the underlying forward elliptic system, which extend results obtained in J. Differential Equations 84 (2): 252-272 for the single elliptic equation and those obtained in Applicable Analysis DOI:10.1080/00036811.2020.1758314, where a Lipschitz type stability estimate of μa on ∂Ω was established in terms of the measurements.